Derivative Notation - AP Calculus BC

1. Find the derivative of $f(x) = \sqrt{x^3}$.

   $f'(x) = \frac{3}{2}x^{1/2}$. Rewrite $\sqrt{x^3} = x^{3/2}$ and use power rule.

2. How is the derivative of $f(x)$ denoted using Newton's notation?

   $\dot{f}(x)$. Dot notation represents time derivative in physics.

3. Find the derivative of $f(x) = \ln(\sin x)$.

   $f'(x) = \cot x$. Chain rule with $\ln(\sin x)$.

4. Find the derivative of $f(x) = \frac{1}{3}x^3$.

   $f'(x) = x^2$. Constant factor $\frac{1}{3}$ remains, apply power rule.

5. What is the derivative notation using Leibniz's notation for $y = f(x)$?

   $\frac{dy}{dx}$. Leibniz notation shows derivative of $y$ with respect to $x$.

6. Find the derivative of $f(x) = 3x^2 + 2x$.

   $f'(x) = 6x + 2$. Apply power rule to each term separately.

7. Find the derivative of $f(x) = 5e^x - 4$.

   $f'(x) = 5e^x$. Constant multiple rule with exponential derivative.

8. State the formula for the derivative of $f(x) = x^n$.

   $f'(x) = nx^{n-1}$. Power rule: bring down exponent, reduce power by 1.

9. What is the derivative of $f(x) = \cot x$?

   $-\csc^2 x$. Derivative of cotangent is negative cosecant squared.

10. Find the derivative of $f(x) = 8x^{-1/2}$.

    $f'(x) = -4x^{-3/2}$. Constant 8 times power rule on $x^{-1/2}$.

11. How is the derivative of $f(x)$ at $x = a$ denoted using prime notation?

    $f'(a)$. Prime notation denotes derivative at specific point.

12. Find the derivative of $f(x) = x^2 \sin x$.

    $2x \sin x + x^2 \cos x$. Apply product rule: $(uv)' = u'v + uv'$.

13. Find the derivative of $f(x) = x^4 - 3x^2 + x$.

    $f'(x) = 4x^3 - 6x + 1$. Apply power rule to each term.

14. Find the derivative of $f(x) = x^3 - 5x + 4$.

    $f'(x) = 3x^2 - 5$. Derivative of constant is 0, apply power rule to other terms.

15. What does the derivative represent geometrically?

    Slope of the tangent line. Derivative gives instantaneous rate of change.

16. What is the derivative of $f(x) = \cot x$?

    $- \csc^2 x$. Derivative of cotangent is negative cosecant squared.

17. What is the derivative of $f(x) = a^x$ where $a > 0$?

    $a^x \ln a$. Exponential with base $a$ requires $\ln a$ factor.

18. What is the derivative of a constant function $f(x) = c$?

    0. Constants have zero rate of change.

19. Find the derivative of $f(x) = 7x^5$.

    $f'(x) = 35x^4$. Constant multiple of power rule.

20. What is the derivative of $f(x) = e^x$?

    $e^x$. Exponential function $e^x$ is its own derivative.

21. Find the derivative of $f(x) = \frac{1}{x}$

    $f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and use power rule.

22. Find the derivative of $f(x) = \frac{1}{2}x^{-2}$.

    $f'(x) = -x^{-3}$. Constant $\frac{1}{2}$ times power rule on $x^{-2}$.

23. Find the derivative of $f(x) = \frac{x^2 + 1}{x}$.

    $f'(x) = \frac{x^2 - 1}{x^2}$. Rewrite as $x + x^{-1}$ and differentiate.

24. Find the derivative of $f(x) = \sqrt{x}$.

    $f'(x) = \frac{1}{2\sqrt{x}}$. Rewrite as $x^{1/2}$ and apply power rule.

25. What is the derivative of $f(x) = \sin x$?

    $\cos x$. Derivative of sine is cosine.

26. Find the derivative of $f(x) = \ln(x^2 + 1)$.

    $\frac{2x}{x^2 + 1}$. Use chain rule with $\ln$ and $x^2 + 1$.

27. Find the derivative of $f(x) = \frac{1}{x^3}$.

    $f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.

28. Find the derivative of $f(x) = 2\sec x$.

    $f'(x) = 2\sec x \tan x$. Constant multiple of secant derivative.

29. What is the derivative of $f(x) = \cos x$?

    $-\sin x$. Derivative of cosine is negative sine.

30. Find the derivative of $f(x) = x \cos x$.

    $f'(x) = \cos x - x \sin x$. Product rule with $u = x$ and $v = \cos x$.

31. What is the derivative of $f(x) = \csc x$?

    $- \csc x \cot x$. Derivative is negative product of cosecant and cotangent.

32. Find the derivative of $f(x) = \frac{1}{x}$.

    $f'(x) = -\frac{1}{x^2}$. Rewrite as $x^{-1}$ and use power rule.

33. Find the derivative of $f(x) = \sqrt{x}$.

    $f'(x) = \frac{1}{2\sqrt{x}}$. Rewrite as $x^{1/2}$ and apply power rule.

34. What is the derivative of $f(x) = \ln x$?

    $\frac{1}{x}$. Natural log derivative is reciprocal function.

35. What is the derivative of $f(x) = \cos x$?

    $-\sin x$. Derivative of cosine is negative sine.

36. Find the derivative of $f(x) = \frac{1}{x^3}$.

    $f'(x) = -\frac{3}{x^4}$. Rewrite as $x^{-3}$ and apply power rule.

37. What is the derivative of $f(x) = \tan x$?

    $\sec^2 x$. Derivative of tangent is secant squared.

38. What is the derivative of $f(x) = \sec x$?

    $\sec x \tan x$. Derivative involves product of secant and tangent.

39. Find the derivative of $f(x) = \tan(x^2)$.

    $f'(x) = 2x \sec^2(x^2)$. Chain rule: outer derivative times inner derivative.

40. Find the derivative of $f(x) = \frac{1}{3}x^3$.

    $f'(x) = x^2$. Constant factor $\frac{1}{3}$ remains, apply power rule.

41. What is the derivative of $f(x) = \sin x$?

    $\cos x$. Derivative of sine is cosine.

42. What is the limit definition of the derivative of $f(x)$ at $x = a$?

    $f'(a) = \lim_{{h \to 0}} \frac{f(a+h) - f(a)}{h}$. Limit of difference quotient as $h$ approaches 0.

43. Find the derivative of $f(x) = \sqrt{x^3}$.

    $f'(x) = \frac{3}{2}x^{1/2}$. Rewrite $\sqrt{x^3} = x^{3/2}$ and use power rule.

44. Find the derivative of $f(x) = \sin^2 x$.

    $f'(x) = 2\sin x \cos x$. Use chain rule with $\sin^2 x = (\sin x)^2$.

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